Question

The number of values of $x$ such that $x,$ $\left[x\right]$ and $\left\{x\right\}$ are in arithmetic progression is equal to (where, $\left[\cdot \right]$ denotes the greatest integer function and $\left\{\cdot \right\}$ denotes the fractional part function)

• A. $0$
• B. $1$
• C. $2$
• D. $4$

Answer 1

Answer: (C)

We have,
$2\left[x\right]=x+\left\{x\right\}=\left[x\right]+2\left\{x\right\}$
$\Rightarrow \left\{x\right\}=\frac{\left[x\right]}{2}.$
Now, $0\le \left\{x\right\}<1$
$\Rightarrow 0\le \frac{\left[x\right]}{2}<1$
$\Rightarrow 0\le \left[x\right]<2\Rightarrow \left[x\right]=0,1$
for $\left[x\right]=0,\left\{x\right\}=0\Rightarrow x=0$
for $\left[x\right]=1,\left\{x\right\}=\frac{1}{2}\Rightarrow x=\frac{3}{2}$
Thus, $x=0$ and $\frac{3}{2}.$